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Theorem ineccnvmo 35063
Description: Equivalence of a double restricted universal quantification and a restricted "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 2-Sep-2021.)
Assertion
Ref Expression
ineccnvmo (∀𝑦𝐵𝑧𝐵 (𝑦 = 𝑧 ∨ ([𝑦]𝐹 ∩ [𝑧]𝐹) = ∅) ↔ ∀𝑥∃*𝑦𝐵 𝑥𝐹𝑦)
Distinct variable groups:   𝑥,𝐵,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧

Proof of Theorem ineccnvmo
StepHypRef Expression
1 relcnv 5807 . . 3 Rel 𝐹
2 id 22 . . . 4 (𝑦 = 𝑧𝑦 = 𝑧)
32inecmo 35061 . . 3 (Rel 𝐹 → (∀𝑦𝐵𝑧𝐵 (𝑦 = 𝑧 ∨ ([𝑦]𝐹 ∩ [𝑧]𝐹) = ∅) ↔ ∀𝑥∃*𝑦𝐵 𝑦𝐹𝑥))
41, 3ax-mp 5 . 2 (∀𝑦𝐵𝑧𝐵 (𝑦 = 𝑧 ∨ ([𝑦]𝐹 ∩ [𝑧]𝐹) = ∅) ↔ ∀𝑥∃*𝑦𝐵 𝑦𝐹𝑥)
5 brcnvg 5600 . . . . 5 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦𝐹𝑥𝑥𝐹𝑦))
65el2v 3422 . . . 4 (𝑦𝐹𝑥𝑥𝐹𝑦)
76rmobii 3336 . . 3 (∃*𝑦𝐵 𝑦𝐹𝑥 ↔ ∃*𝑦𝐵 𝑥𝐹𝑦)
87albii 1782 . 2 (∀𝑥∃*𝑦𝐵 𝑦𝐹𝑥 ↔ ∀𝑥∃*𝑦𝐵 𝑥𝐹𝑦)
94, 8bitri 267 1 (∀𝑦𝐵𝑧𝐵 (𝑦 = 𝑧 ∨ ([𝑦]𝐹 ∩ [𝑧]𝐹) = ∅) ↔ ∀𝑥∃*𝑦𝐵 𝑥𝐹𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wo 833  wal 1505   = wceq 1507  wral 3088  ∃*wrmo 3091  Vcvv 3415  cin 3828  c0 4178   class class class wbr 4929  ccnv 5406  Rel wrel 5412  [cec 8087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-br 4930  df-opab 4992  df-xp 5413  df-rel 5414  df-cnv 5415  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-ec 8091
This theorem is referenced by:  ineccnvmo2  35066
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